Suppose that a contractor employs carpenters, bricklayers, and plumbers, working

Question

Suppose that a contractor employs carpenters, bricklayers, and plumbers, working three shifts per day. The number of labor-hours employed in each of the shifts is summarized in the following matrix:

Shift
1 2 3
[ 50 20 10 ] Carpenters
[ 30 30 15 ] Bricklayers
[ 20 20 5 ] Plumbers

Labor in shift 1 costs $10 per hour, in shift 2 $15 per hour, and in shift 3 $20 per hour. Determine a matrix calculation resulting in a matrix having three entries, where the first entry is the amount spent on carpenters, the second entry is the amount spent on bricklayers, and the third entry is the amount spent on plumbers.

State the matrix calculation, how the calculation is carried out, and the entries in the resulting matrix.

Explanation / Answer

The given matrix represents the type of workers and number of workers in each shift. The rows represent the type of worker, and the columns represent the shift. You want to multiple this matrix with a 3x1 matrix that represents the cost per hour at each shift, like the following: [10]
[15] [20]
Multiplying this new matrix with the original matrix gives us the following: [50 20 10]    [10]    [50x10 + 20x15 + 10x20]    [500+300+200]    [1000]
[30 30 15] x [15] = [30x10 + 30x15 + 15x20] = [300+450+300] = [1050]
[20 20 5]     [20]     [20x10 + 20x15 + 5x20]     [200+300+100]     [600]
where 1000 is the amount spent on carpenters, 1050 is the amount spent on bricklayers, and 600 is the amount spent on plumbers.

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